Question: If the 25th day of the year 2003 falls on a Saturday, on what day of the week did the 284th day of the year 2003 fall?
Answer: There are 7 days in a week.  Two days of the year fall on the same day of the week if and only if they are congruent modulo 7.  Notice that  \[284\equiv4\pmod7\] and  \[25\equiv4\pmod7.\] Therefore the 284th day and the 25th day fall on the same day of the week.  Therefore the 284th day of the year falls on a $\boxed{\text{Saturday}}$.